The song is usually known as Rock All Our Babies to Sleep . . .

Lyrics and brief discussion in this thread.

Lots of people have done it, I'll up some more stuff later today.

In the meantime, here's the Ballad Index entry . . .

Rocking the Cradle (and the Child Not His Own)
DESCRIPTION: The old man laments "about rocking the cradle and the child not his own." Though at the time he had been happy to marry a lighthearted lass, he now finds her out at parties all the time (or keeping company with other men)
AUTHOR: unknown
EARLIEST DATE: 1924 (recording, Riley Puckett)
KEYWORDS: marriage age wife husband children infidelity bastard
FOUND IN: Ireland US(SE,So) Britain(Wales) Canada(Newf) Australia
REFERENCES (7 citations):
Randolph 393, "Rock All Our Babies to Sleep" (1 text, 1 tune)
Warner 166, "Show Me the Man Who Never Done Wrong (or, Rocking the Baby to Sleep)" (1 text, 1 tune -- a curious version in which it appears at first that it is the woman, not the man, who is betrayed)
Meredith/Anderson, pp. 168-169, "The Wee One"; p. 266, "Rock All Our Babies" (2 texts, 2 tunes)
Kennedy 212, "Rocking the Cradle" (1 text, 1 tune)
Peacock, pp. 478-479, "The Milkman's Lament" (1 text, 1 tune)
Lomax-FSNA 192, "The Old Man's Lament" (1 text, 1 tune); also 190, "Run Along, You Little Dogies" (1 text, 1 tune, mostly "Get Along Little Dogies" but with a chorus partly from this piece!)
DT, ROCKCRAD ROCKCRA2
Roud #357
RECORDINGS:
A. L. Lloyd, "Rocking the Cradle" (on Lloyd2, Lloyd4)
Uncle Dave Macon, "Tossing the Baby So High" (Vocalion 5013, 1926)
Neil Morris, "Rock All the Babies to Sleep" (on LomaxCD1707)
Charlie & Bud Newman, "Rock All Our Babies to Sleep" (OKeh 45431, 1930; rec. 1928)
Riley Puckett, "Rock All Our Babies to Sleep" (Columbia 107-D, 1924)
George Reneau, "Rock All Our Babies to Sleep" (Vocalion 14997, 1925)
Jimmie Rodgers, "Rock All Our Babies to Sleep" (Victor 23721, 1932; Regal Zonophone [UK] MR-2200, 1936; rec. 1930)
Dave Turner [pseud. for Dick Parman], "Rock All Our Babies To Sleep" (Supertone 9374, 1929)
Fay & Jay Walker, "Rock All Our Babies to Sleep" (Broadway 8093, c. 1925)
CROSS-REFERENCES:
cf. "Unhappy Jeremiah (The Brats of Jeremiah)" (plot)
cf. "Hush-a-Bye, Baby" (plot)
cf. "When I Was Single (II)"
ALTERNATE TITLES:
Tossing the Baby So High (Uncle Dave Macon version)
Notes: An Irish legend has it that the chorus, "Hi-ho, hi-ho, my laddie, lie easy, For perhaps your own daddy might never be known. I'm seein' and sighin' and rockin' the cradle, And nursing the baby that's none of my own," was sung by the Virgin Mary to the baby Jesus. In English, no doubt. - RBW